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Based on a vector-bundle formulation, we introduce a new family of nonlinear subdivision schemes for manifold-valued data. Any such nonlinear subdivision scheme is based on an underlying linear subdivision scheme. We show that if the underlying linear subdivision scheme reproduces Π k , then the nonlinear scheme satisfies an order k proximity condition with the linear scheme. We also develop a new "proximity ⇒ smoothness" theorem, improving the one in [12]. Combining the two results, we can conclude that if the underlying linear scheme is C k and stable, the nonlinear scheme is also C k .The family of manifold-valued data subdivision scheme introduced in this paper includes a variant of the log-exp scheme, proposed in [10], as a special case, but not the original log-exp scheme when the underlying linear scheme is non-interpolatory. The original log-exp scheme uses the same tangent plane for both the odd and the even rules, while the variant uses two different, judiciously chosen, tangent planes. We also present computational experiments that indicate that the original smoothness equivalence conjecture posted in [10] is unlikely to be true.Our result also generalizes the recent results in [17,16,5,6]. It uses only the intrinsic smoothness structure of the manifold and (hence) does not rely on any embedding or Lie group or symmetric space or Riemannian structure. In particular, concepts such as geodesics, log and exp maps, or projection from ambient space play no explicit role in the theorem. Also, the underlying linear scheme needs not be interpolatory.Here (a ) and (b ) comes from the mask of a linear subdivision scheme T . It was conjectured by Donoho that this scheme satisfies a so-called smoothness equivalence property: if T produces C k smooth curves, then so does S. While this conjecture had stimulated a number of studies and partial solutions, e.g. [17,16,5,6] and the references therein, the conjecture remains unsolved.

Interpolation of manifold-valued data is a fundamental problem which has applications in many fields. The linear subdivision method is an efficient and well-studied method for interpolating or approximating real-valued data in a multiresolution fashion. A natural way to apply a linear subdivision scheme S to interpolate manifold-valued data is to first embed the manifold at hand to an Euclidean space and construct a projection operator P that maps points from the ambient space to a closest point on the embedded surface, and then consider the nonlinear subdivision operator S := P • S. When applied to symmetric spaces such as S n−1 , SO(n), SL(n), SE(n), G(n, k) the projection method can also be carried out in such a way that the resulted schemes enjoy natural coordinate invariance properties and robust numerical implementations. Despite such nice features, the mathematical analysis of such nonlinear subdivision schemes is at its infancy. In this article, we attack the so-called Smoothness Equivalence Conjecture, which asserts that the smoothness property of S is exactly the same as that of S. We show that in the cases of S n−1 , SO(n) and related manifolds, we have a proximity condition of the form: (S − S)y ∞ p−1 i=1 |∆ i y| ∞ |∆ p−i y| ∞ , where p is the accuracy order of S. Armed with this proximity condition and other known approximation theoretic results, we can establish the result that the Hölder smoothness exponent of S is always as high as that of S-no matter how high the latter is.

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